On the parabolic Fatou domains II: rigidity

This paper is a follow-up study on the holomorphic model problem for infinitely-connected parabolic Fatou domains of rational maps. We prove that simple parabolic maps serve as holomorphic models for such parabolic Fatou domains. Moreover, we show that every simple parabolic map can be perturbed into a rational map with a completely invariant attracting Fatou domain without changing the topology of the Julia set, thereby confirming the Goldberg–Milnor conjecture for simple parabolic maps.